Introduction to Dice Stats

now, we want to calculate 2 arrays, the first is the array with the list of all possible throw in order: lets call this a1
it start at a+c and finishes at ab+c
so

a+c, a+c+1, ..., ab+c-1, ab+c

how long is this array ? ab-a+1

now let's consider a second array (a2), as long as the first, and fill it with the corresponding values of the probability of the throw.
We discovered that we can calculate the values of the probability from the diagonals of the Pascal's triangle.
the values are exactly the values of the diagonal number a, up to the value b
The additional values, up to the half of the array lenght or half of the array leght +1 (if the array is odd) are calculated as : number of the pascal's triangle-(number of the 1st pascal triangle at the same diagonal*a), number of the pascal's triagle-(number of 2nd pascal triangle at the same diagonal*a)

the a (so the third) pascal's triangle diagonal is: 1,3,6,10,15,21,28,36,….
so let's put the numbers of the diagonal up to b in the a2
a2=1,3,6,10,15,x,y,x,15,10,6,3,1
x will be 21-(a*1)=18
where 1 is the first number of the same diagonal
y will be 28-(a*3)=19
where 3 is the second number of the same diagonal
as a check of our calculations: 1+3+6+10+15+18+19+18+15+10+6+3+1=125
and 125 = 5^3 (b^a)

The laser resistance by level is in the hpBonus terminology used in most places in the game data

Level

1

2

3

4

5

6

7

8

9

10

Armor Laser Resistance

+0

+25

+67

+150

+300

+614

+1150

+2400

+4900

***

The formula in question is 100 / (100 + hpBonus) = percentile_resistance = 1 - damage_adjustment. Remembering that percent is literally per hundred and to get a real ratio you simply move the decimal two places more significant.

First let's look at the laser colimnator. It adds +75% damage.

The laser is easy. Just 1d4.

possible rolls

1

2

3

4

average

laser

1

2

3

4

2.5

colimnated laser

1

3

5

7

4

As you can see it only actually gains +60%.

The turbolaser is more troublesome with multiple dice.

possible rolls

3

4

5

6

7

8

9

10

11

12

average

probability

1/64

3/64

3/32

5/32

3/16

3/16

5/32

3/32

3/64

1/64

-

turbolaser

3

4

5

6

7

8

9

10

11

12

7.5

colimnated turbolaser

5

7

8

10

12

14

15

17

19

21

12.75

Now we're getting +70%.

Of course doing that in a table is cumbersome. Unfortunately you cannot algebraicly simplify a series that contains the floor function. Machine assistance is suggested.

Because I don't have anything like that set up right now I'm only going to do one example. Well, two but since level 1 armor takes full rated damage from lasers that one's trivial.

possible rolls

1

2

3

4

average

laser raw damage

1

2

3

4

2.5

against light titanium

1

2

3

4

2.5

against light plasteel

0

1

1

2

1

Rather than doing 60% as much damage against level 3 armor as against level 1 the laser is doing 40%, a 50% drop in effectiveness due to roundoff.

possible rolls

3

4

5

6

7

8

9

10

11

12

average

probability

1/64

3/64

3/32

5/32

3/16

3/16

5/32

3/32

3/64

1/64

-

turbolaser raw damage

3

4

5

6

7

8

9

10

11

12

7.5

against light titanium

3

4

5

6

7

8

9

10

11

12

7.5

against light plasteel

1

2

3

3

4

4

5

6

6

7

4.09375

Roundoff still hurts dropping the damage from 60% to just under 55%, but that's about a 9% reduction in effectiveness, much better than the 50% drop the laser suffers from theoretical against the same armor.